Integration by trigonometric substitution

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Sometimes directly can not perform the integrals, at other times it seems that we could integrate immediately because first inspection to find similarity with the formulas that have formulas in tables. There are even some of the same formulas we can deduce by some techniques, like the one in this case in point, consider the following example: Deduce the following formula:

 

Think we can make a substitution in the integral in such a way that allows us immediate integration. Recall that:

note that if we make a change of variables that will lead to the use of this substitution, namely, we replace

  

Recall that it is also expressed as:

           

where

where c is a new constant generated coupled with the logarithm:

like this integral can be found in the same way some others, it is worth continuing the following recommendation:

  

We clarify that these substitutions emerge as the replacement of the previous year of observation and comparison of trigonometric properties:

Calculate and check the following integral

   

Solution:

as we can see the integration can not be performed immediately. Before performing any replacement would be worth making some radical factorization

performing substitution

 

therefore:

and then:

identify the right triangle below:

the hypotenuse is 2x and the adjacent side is 3 so the opposite side is equal to:

so that

Check the results.

simplifying we have:

They suggest the following exercises:

Trigonometric substitution

It is often possible to find the antiderivative of a function when the integrand presents expressions of the form:

Radical is removed by the appropriate trigonometric substitution, the result is an integrand containing trigonometric functions whose integration is familiar. The following table shows what should be the replacement:S o l u c i o n e s

 

Substituting these values into (1) yields:

 

Substituting these values into (1) yields:

 

 

 

 

Integration by trigonometric substitution

Substitutions involving trigonometric functions can be performed in those integrals whose integrating contains an expression of the form:

with $ 0 “> y $ 0″>

The trigonometric substitution integral transforms one another that contains trigonometric functions whose integration is easier.

Study each case as follows:

A. The integrand contains a form with function 0 \,, \, b> 0 $ “>

It makes the change of variable typing

where

If then

Furthermore:

as $ 0 “> and as

then $ 0 “> so

Then:

Since then

For this case, the other trigonometric functions can be obtained from the following figure:

Examples:

Be with

Then:

Substituting:

Since then and

Furthermore therefore

These results can also be obtained from the following figure:

Finally:

Sea

Then

Substituting

As then so you can use the following figure to give the final result:

Then:

Sea

Furthermore:

Substituting:

Sea

Then

Substituting

and therefore

It can also be used:

B) The integrand contains an expression of the form with 0 \,, \, b> 0 $ “>

We make a change of variable typing wherever and

If then

Also

As 0} $ “> and then is positive

and therefore

The other trigonometric functions can be obtained from the following figure:

Examples:

Sea

Then:

Substituting

Sea

Then:

0 if \ theta \ varepsilon \ left] \ frac {- \ pi} {2}, \ frac {\ pi} {2} \ right [\ right)} $ “>

Substituting

Sea

Then

Substituting

As

Therefore:

Sea

Then

Substituting

Since then

Therefore:

Finally:

c.

The integrand contains an expression of the form with $ 0 “> y $ 0″>

In this case the appropriate substitution is:

where

and \ frac {a} {b}} $ “>

If then

Also

where

as $ 0 “> y $ 0″> to

As so then

Using the triangle can be obtained following the other trigonometric functions:

Examples:

Sea

Then

Substituting:

Sea

Then

Substituting:

Sea

Then

Substituting:

As the following figure can be used to determine

Finally:

Other integrals used in any of the trigonometric substitutions that we have studied are those containing an expression of the form. The following examples illustrate the procedure:

Examples:

Ie we can write as

It is then that we must calculate the integral

Sea

Then

Substituting:

It must:

Then the integral becomes:

and replacing used where:

Then:

Substituting:

with or

Must

therefore, with $ 1 “>

wherein either

Then and

Substituting

Must (completing squares)

Then the integral is to be determined:

Sea

Then

Substituting

Since then, using that

finally obtained

with

In each case determine the range over which the result is valid.